# "conjugate with" or "conjugate to"

This is a question specifically about English language usage in mathematics, but it is probably the same in all topics (however I can't really think of an example in everyday life!).

Should I say "A is conjugate to B" or "A is conjugate with B"? I'm pretty sure "A is the conjugate of B" is acceptable, but I think (at least one) of the previous is also correct.

I'm referring to "conjugate" as in "subgroups are conjugate".

• Try "a and b are conjugate." You seem to be implying some sort of unique conjugacy, perhaps with the conjugacy operator fixed. "a and b are congugate" just means that some operator does exist which satisfies the conjugacy condition. Nov 15, 2016 at 22:54
• In the phrase "subgroups are conjugate", it would also be common to make the subject and object agree in number and say "subgroups are conjugates" (with an implied "of each other") although the meaning is slightly different than the singular "conjugate" which refers to the relationship itself rather than the relationship of members of the set whose members are conjugate. Nov 16, 2016 at 6:39

As per Wikipedia on algebraic conjugates, it would be correct to say, "A is the conjugate of B," as you have noted. And of course, B is the conjugate of A, given that conjugates come in pairs.

Based on my experience, it would also be correct to say, "A is conjugate to B," with the reverse also being true. No mathematician's eyebrows would go up.

On the other hand, I don't know anyone who deals with these issues on a regular basis who would say, "A is conjugate with B." Instead, they would say, "A and B are conjugates," as suggested by @PhilSweet. I admit that this is on the soft side for ELU, but that's just the way it is.

• As an undergraduate math major, I would agree that Richard Kayser is correct. Using the preposition "of", or dispensing with a preposition would be preferred, but the preposition "to" would not be incorrect, while the preposition "with" would be incorrect in most circumstances. Nov 16, 2016 at 6:36
• @ohwilleke Thanks for the confirmation. I took enough math courses in college to major in math, and I still like to work problems in number theory for fun, but it's hard to explain why "to" is acceptable while "with" is not, at least to the standards of ELU. Good to know you're a math major. Stick with it. Nov 16, 2016 at 6:41
• Kayser I was a math major; now I'm a middle aged lawyer. In general, the appropriate preposition in a particular context is arbitrary and incidentally frequently does not translate to other languages well as other languages have made alternate arbitrary choices as they have arisen. Nov 16, 2016 at 6:50
• @ohwilleke Agree. I guess you didn't stick with it. :-) But law has its logic, despite what some might say to the contrary. Nov 16, 2016 at 6:58
• The common failing of young lawyers is to overestimate how logical law is, the life of law is not reason, it is experience. Nov 16, 2016 at 7:00